# On the second eigenvalue of random bipartite biregular graphs

@article{Zhu2020OnTS, title={On the second eigenvalue of random bipartite biregular graphs}, author={Yizhe Zhu}, journal={arXiv: Probability}, year={2020} }

We consider the spectral gap of a uniformly chosen random $(d_1,d_2)$-biregular bipartite graph $G$ with $|V_1|=n, |V_2|=m$, where $d_1,d_2$ could possibly grow with $n$ and $m$. Let $A$ be the adjacency matrix of $G$. Under the assumption that $d_1\geq d_2$ and $d_2=O(n^{2/3}),$ we show that $\lambda_2(A)=O(\sqrt{d_1})$ with high probability. As a corollary, combining the results from Tikhomirov and Youssef (2019), we confirm a conjecture in Cook (2017) that the second singular value of a…

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